# Homotopy Type Theory Heyting division Z-algebra > history (changes)

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## Definition

A

Heyting division $\mathbb{Z}$-algebra is a $\mathbb{Z}$-algebra $(A, +, -, 0, \cdot)$ with

• a tight apartness relation type family $a # b$ for $a:A$, $b:A$
• a term showing that all endofunctions of $A$ are strongly extensional
$s:\prod_{(f:A \to A)} \prod_{(a:A)} \prod_{(b:A)} (a # b) \to (f(a) # f(b))$
• a left divisibility identity
$d_\lambda:\prod_{(a:A)} \left( (a # 0) \times \prod_{(c:A)} \left\Vert \sum_{(b:A)} a \cdot b = c \right\Vert \right)$
• a right divisibility identity
$d_\lambda:\prod_{(a:A)} \left( (a # 0) \times \prod_{(c:A)} \left\Vert \sum_{(b:A)} b \cdot a = c \right\Vert \right)$

## Examples

• The rational numbers are a Heyting division $\mathbb{Z}$-algebra.

• Every discrete division Z-algebra is a Heyting division $\mathbb{Z}$-algebra.

• Every Heyting division ring is a Heyting division $\mathbb{Z}$-algebra.